{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "# Rendering math equations using TeX\n",
    "\n",
    "\n",
    "You can use TeX to render all of your Matplotlib text by setting\n",
    ":rc:`text.usetex` to True.  This requires that you have TeX and the other\n",
    "dependencies described in the :doc:`/tutorials/text/usetex` tutorial properly\n",
    "installed on your system.  Matplotlib caches processed TeX expressions, so that\n",
    "only the first occurrence of an expression triggers a TeX compilation. Later\n",
    "occurrences reuse the rendered image from the cache and are thus faster.\n",
    "\n",
    "Unicode input is supported, e.g. for the y-axis label in this example.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Duplicate key in file WindowsPath('e:/ruanjian/python38/lib/site-packages/matplotlib/mpl-data/matplotlibrc'), line 770 ('font.family        : sans-serif       ')\n",
      "Duplicate key in file WindowsPath('e:/ruanjian/python38/lib/site-packages/matplotlib/mpl-data/matplotlibrc'), line 771 ('font.sans-serif    : SimHei, Bitstream Vera Sans, Lucida Grande, Verdana, Geneva, Lucid, Arial, Helvetica, Avant Garde, sans-serif ')\n",
      "Duplicate key in file WindowsPath('e:/ruanjian/python38/lib/site-packages/matplotlib/mpl-data/matplotlibrc'), line 772 ('axes.unicode_minus:False')\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, '\\\\TeX\\\\ is Number $\\\\displaystyle\\\\sum_{n=1}^\\\\infty\\\\frac{-e^{i\\\\pi}}{2^n}$!')"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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+3od/37gYoUF6tUMCwAlqUp57MBPTDKH48Z6T6B/kEwEbW+/AEP757dMwxoXj+/ekue04U4ID8NyDmai81oEXv7jstuMw/7H/TAPeO3kN/5A/BwunG9QO5wZOUJMQERKInz40H5eau/Dy4Wq1w2Fe7r8/voia1m78r0fmIzjAvVeo92UlYfW8BPy65DzqzD1uPRbzbf2DFvzbh5WYnTAFW+4yjv8CD+IENUn3zI3Hnelx+N8fXUB7N1dCZyO72NiJ335ahUeXJOOOtDi3H4+I8JOHsiCEMteKsdH84XA1Lrd049n7M1w2YMdVvCsaH0REeHZdBtp7BvBfH/OACTayn75XibCgADx7v+duPk+PDsPf581GcUUDPr/Q5LHjMt/R3j2A/33wAlbOjsM9c+LVDucWnKBcIHNaJNbnTMdLX1ajpqVb7XCYlymrbsWn55vw/XvSEDcl2KPH/vaKmZgWFYJfFZ/ntaTYLf7zowvo6B3As+syvLLGKCcoF/lRwVzodcTzT9gtfl18AXFTgvDk8olXi5io4AA9/nrVbByrMeOT89yKYsOqW7rw0leXsT5nOjKmemd5LE5QLpIYGYLNdxnx/qlrOFFrVjsc5iW+NrXg0MVmbL07DWFBaiy/BqzPnY6UmFD8mltRzMYvD5xHgE6HHxXMVTuUUXGCcqFNdxkRGRKA33wy0nqMTGuEEPhl8XnERwTjW5OotTdZgXod/mbVbJy80o6SykbV4mDeo7qlC++dvIonlqciMTJE7XBGxQnKhaYEB+DJ5TOxv6IeFxu5Tp/WfVnVgiOXWvHDe9IQEqjuxMdHlyQjNTYMvyo+D4uFW1Fat+szEwJ0OnzvzllqhzImjyUoIjIQUTYRFcrFC+33ZxNRFRGVya8iub2NiIqJaJunYp2M76yYieAAHXZ9xq0oLRNC4FfF5zE1KgTfvG2G2uEgQK/D362ejcprHdjP60ZpWmNnL/aUXcFjOdO9uvUEeLYFtQFArnWZdyLabLc/RgiRJoTIAbAJygKHALBeCJEvhNjhwVgnLHZKMDbmpuCtY3W41s4TJLWqvKYNZdVt2Hq3+q0nq4cWJ2NmbBh2fmZSOxSmohe/uIzBIYvXTcodiccSlBBil3UVXSjLt5vs9tuulmsUQlj3G4jI+z9JG0+tNMIigBc+v6R2KEwlvzt0GZEhASjMma52KDfodYTvrJiF47VmlHMhWU3q6B3Ay19VY+2CqZgZF652OOPy+D0omWxaR1u+nYg2W1tZUgyAViLaOdrziaiUiEqbmrxjGG1KTBgeXDQNfzpSw4vHadCVtm58ePoaHr9tBsKD1Rm5N5rCnOmICAnA7w7xxZMWvXy4Gp19g/j+3e6rBelKagySKBRCbBljf77tA9nyMgMwE1Gh/ZPl/lwhRG58vPfMhN56dxq6+4fwB67Rpzl/+KoaRIQn75ipdii3CA8OwDeXpuDD0/W4yjX6NKVvcAi/O3QZK2fHYX5ylNrhOMSjCYqICq33kogoe4T9BrvHm0d6ni+YmxSBu+bE449f1/CSBxrS3T+IPx2pwZqsJCQbQtUOZ0RPLp8JIQR+/xVfPGnJvtP1aL7eh6dW+s4dE0+O4ssDUGQdpQel6w5EVGzztBgAtosrvS6fUwgAdl1/Xu9bt89AfUcvzz3RkDfK69DRO4jv3jlT7VBGlRIThvuykvCnIzXo6R9SOxzmIS8frkZqbBhWpru/WLGreHKQRIl1lJ78KpHb822eY7Lt/hNCmIUQ5UKIvUKI7Z6K1VVWzUvA1KgQvPI1X6lqgcUi8OIXl7BoehSyZ0SrHc6YvnvnLLT3DODNY1fUDoV5wNn6Dhy93Ia/vH0GdDrvq7k3Gp6o60YBeh0ev20GPr/QjEvNXWqHw9zsswtNMDV14bt3zvLKwpu2clOjsSA5Ci9+cZnLH2nAK4drEBSgw/qcFLVDcQonKDf75tIUBOgIr/BgCb/36pFaxIYHYe38qWqHMi4iwhPLU3Gx8TrKqnnIuT+73jeIN8uv4IGFUxEdHqR2OE7hBOVmCZEhuC8rCXvKrqB3gPv7/VVTZx9KKhvwaHYyggJ848/q/gVTER6kx6tHa9UOhbnR28fq0NU/pGo9yInyjb8kH/eXy2agvWcA7528pnYozE3eOnYFgxaBjUt9pwslPDgADy6ehvdPXkNnL68G7Y+EEHj5cDUyp0ZiSYpB7XCcxgnKA5YbY5EWH46XuZvPLwkh8OrRWuSkRiM9IULtcJyyITcFPQNDePcEXzz5o/KaNpyt78S3lqV6/X3RkXCC8gAiwuO3zcDxWjMuNHSqHQ5zsbLqNpiaunyq9WS1OMWAuYkReK2Uu/n80etHryAsSI+HFk9TO5QJ4QTlIQ8tToZeR3ijvE7tUJiLvXq0FuFBety/wPsHR9gjImxYmoITtWZUXutQOxzmQr0DQ3j/1DWsnT/V60puOYoTlIfERwTjnjnxeOvYFQzxejx+o7N3AO+fvIYHF0/z2ZPAI0uSEaTX4TUeLOFX9p+px/W+QTyWk6x2KBPGCcqDHsuZjoaOPnxxsVntUJiLvHviGnoGhrBxqfprPk1UTHgQCrIS8fbxOh5p6kfeKK9DsiEUy2bFqh3KhHGC8qDVGQmIDAnAG+U8e99f7CmrxdzECCya7hvFN0ezcWkKzN0DOMhlufxCQ0cvDl1owiNLkn2qcoQ9TlAeFBygxzcWTcP+M/U8rNcPVLd04ViNGY9kJ/vkCClbd6TFISEiGG8f53uk/uCtY3WwCODRbN/t3gOcSFBEFElEke4MRgsey5mO3gELPjjFw3p93Z+PXwUR8OAi3xwhZUuvIzy4aBo+OdcIczevYebLhBB4o+wKsmcYYIyfonY4kzJqgpIJ6R/lYoBDANoAtBHREBFdIKLXiOhez4XqH5akGGCMC8cbZXyl6suEEHj7eB1umxmDaV66rIazHl6SjIEhgQ9O1asdCpuEU3XtuNB4HY950WrOEzVWC+ojKMtfbAeQLr9PB5AL4GkoCesNmax+5O5A/QUR4bGc6ThyuRU1Ld1qh8Mm6HRdB0xNXXh4iW93odjKmhaJtPhw7ubzcW+W1yEoQIcHFvp+y37UBCVXqX1GCHFQCHFJCNEu/z0mhHhDCLFVCBED4PsAHpeJatFo70dEBiLKJqJCIioa5TltRFRMRNtsthUSUR4RbZ7MD+pNrCe1d09eVTkSNlFvH69DoJ6wzgcKwzqKiPDw4mQcudSKOl5t1ycNDlnw3smryMtIQFRooNrhTNqkB0nIdZ5yAWyE0qL63ihP3QAg17ro4CgJZ70QIt9m1V3rQoUl8nHeZOP1BsmGUOSmRuPdE5ygfNGQReDdE1dxz9wERIX5/knA1oOy4sA7x/l30xcdNrWi+Xq/X9wXBZwbJPE8ER0dbb9cWDAdwGwiWjXC/l1CiF3yoRGAaYS3MRCR7XrES22eZwLgk8u/j+SBhVNxtr6TSx/5oMOmFjR29uHhxf7TvWeVGhuOJTMM+DN38/mkd09cRXiQHvfMTVA7FJdwKEER0W+h3Isy2WxbQkQt9i0mIcTTQoiPxngvI4BWa6vITgyAViLaKR8b7PbfMuOMiDbLgRylTU1Njvw4XmHdwqnQEfAuVzj3OW8fq8OU4ACszvCPk4C9hxcn42x9J87V88WTL+kftGDfmXoUZCUhJFCvdjgu4WgLagOAbUKIjdYNQohjAPYC2OXkaL5C22XdbclWlhmAWXbvmaEkrVHJ1+QKIXLj4+OdCENdCREhWGaMxXsnrvKKpj6kd2AI+07XY818/zkJ2Lt/4VTodcSDJXzMoYtNaO8ZwDcW+c99UWfuQd3SJScTzS8A7HDkDYio0Ob+Urbdvs322wAcxXArygig2Il4vd4DC6fB1NyFCi7S6TM+O9+Ezr5Bv+njH0nclGCsSI/D+yev8cWTD3nvxDVEhQbiznTfuVAfj6MJqgRKK2okB6AkjzHJAQ5FRFRGRGWQLSMisiad1+Vj68CIvXJAhVG+1jBKt6DPWjM/CQE64rV4fMgHp67BEBaI5Wm+W9/MEfcvSEJNazfOXOWLJ1/QOzCEAxUNWJOV5DMrOjvC0Z/kaQAbiOhVIrJfkS3fkTeQo/3ShBA58qtEbs+X/5rlQIu9QojtNq/bIV/rUCvNl8SEB+HO2XF47yR38/mC3oEhlFQ24r7MJATq/eckMJKCzCTodYT3ueKJT/jkXCOu9w3iG37Wsnfor0wIYQJQIL/MRHRUVpK4AOAfATzvxhj92gMLp+FKWw+O15rVDoWN4/MLzbjeN4h1C/2nj3800eFBuCMtFh+c4m4+X/DuyWuIDQ/CMuOYt+x9jsOXgbIVEwNlvlMZgGgAlwBsFUL8wk3x+b2CrEQE6XXczecDrN17d/h5957V/QumorqFu/m8XXf/IA5WNmDdgqkI8LOWvVM/DRHNBFAqq0gUyK/d7glNGyJDAnHXnHjsO81Xqt6sb3AIJRUNKMhM9PvuPauCLKWbjwsbe7ePzzahd8CC+/2wZe/oPKgoOUm3CkAVEQ3K7YvdGJtmrJ2fhKvtvThxpV3tUNgoPj/fjM6+QazzwWXdJyqGu/l8woenle69pTP9q3sPcLwFtRtAGpTBEhsBWBe/eYGI9rkjMC3Jy0hEgI6w7zRXkfZWH5xShvCuSI9TOxSPWrdgKi63dPNUCC/VOzCEj882oiArEXofXphwNI4mqDwATwkhfg7l/pPVBgC3cTXzyYmSw5a5m8879Q0OoVhj3XtW93E3n1c7dKEZXf1DWONHRYttOfrX1grgljOnHN33bwC2ujIoLVo7X7lSPcvlZbzOoQuye88P+/jHExMehOXGWHxwqp4vnrzQh6frERkSgOVG/xy442iCOgjg2VH2lcGBibpsbPmZiSACd/N5oX2n6xEREoAVadrq3rNauyAJl5q7cL7hutqhMBsDQxaUVDYgLyPRrybn2nL0p9oGIIaIjgBYYrdvPUauTM6cEB8RjKUzYzhBeZlBeRJYPS/Bb08C47FePO0/w7+b3uSwqQXtPQNYMz9J7VDcxtGJuu1Q7kO1QykQS3Ky7gUAm6BUOmeTtHZ+Es41dMLUxFeq3uLo5Ta0dQ/gviz/PQmMJyEiBNkzojlBeZkPT9cjLEiPu+b4T+09e85M1L0kyxKlAdgCpWvvDSiLEL7ppvg0xXoS/JBbUV5j/5l6BAfocPdc/z0JOOK+rEScudqB2tZutUNhUBbNPHCmHvfOTfDbqvqAcwsWziSiSJmodsvJuk/LZTeYC0wzhGJRioGvVL2EEALFFQ1YOTsOYUEBaoejqoJM5eLpQEWDypEwACirbkPz9X6/7t4DHJ+ouwnKJN0i+XimrMW338m1oNg41s5Pwskr7agz96gdiuadrutAnbkHBRru3rOaGReOuYkRfPHkJfadrkdQgA73zvPPRTOtHG1BbYfSnWe917QXShVzAlDCScp1CjITAQAlfKWquv1n6qEjZSI1U7r5Si+3ouV6n9qhaJoQAsWV9bgzPQ5Tgv27Ze9ogjIC+K0QooOIogBkQ5m4WwBlwcKnx3sDIjIQUTYRFRJRkaP7iaiNiIqJaJuDsfo0Y/wUpMWH40AFX6mqbf+Zetw2KwYx4UFqh+IVCrKSYBFASSVfPKnpbH0nalt7kJ/p/xdOjiYoE4BZ8vs8KJN2rYsHHgCQ68B7bIAyoGIvoKyg6+D+9UKIfH9cD2o0BVlJ+NrUivbuAbVD0SxT03VcaLyu6dF79rKmRSLZEIr9ZzhBqam4ogFEwOoM/+7eAxxPUDsA7CCiH0OpHFEuhLAW5zJieFn2UQkhdgkhdtm8xuTgfgMRaWoicH5mIgYtAh+fa1Q7FM2ynoT5/tMwIsJ9WUk4JNfFYuo4UFGPJSkGJESEqB2K242aoOQgiFRASR5QBkg8C2WY+Xb5nKcgE5ajB5TJpnW05dtH2B8DoJWIdo7y/M1EVEpEpU1NTY6G4dUWTzcgPiIYxXwfSjUHKuoxP1lpMbBh92Ulon/Igk/44kkVV809OF3XoZkLp7FaUEUAdhPRb+Tw8h1CiBghhE4I8ZF8ztNQBkpscuKYhUKILY7uly0rM5SVfAvtnyz35wohcuPj/WOuik5HyMtIxCfnGtE3OKR2OJrT2NmL47XmG0Or2bCc1GhEhwXyIB6VWC9atXD/CRgjQQkhyuUgiL0APiKi54kowu456TJpHXfkYERUaL2XRETZ4+2XraNbnqcFBZmJ6OofwpdVLWqHojkfVTZCCO2cBJwRoNdh1bxEfHS2EQNDFrXD0ZziigYY48ORFj9F7VA8wpF7UFUAngdwBcOJKtLZAxFRHoAiIiojojIoXXcgouIx9r8u9xUCgHUAhRYsT4tFeJCeu/lUUFzRgGRDKOYlRYz/ZA3Kz0xAR+8gjl5uVTsUTWnvGcBhU4umWvZjJig5Qdc65HsBgJ0ASqEkqh87cyAhRIkQIk0IkSO/SuT2/NH2CyHMsiW3VwihqXp/IYF63D03HsUVDbBYeJkDT+nuH8Shi82yQKr/LQDnCitnxyMoQIeSCr4P5UmfnGvEoEVoqmU/XguqUAixUQjxhhBiK5Qh328IIXIBtMvBCU4lKua4gswkNHX24cQVs9qhaMahC83oG7Ro6iTgrPDgAKxIi0VxJa8R5UkHKhoQNyUYS1IMaofiMeMlKPtLyBuPZT2+XAAdRHTU5ZEx3Ds3AXodcf0zDyquaEBESABumxWjdiheLT8zCbWtPbxGlIf0DQ7h03NNyMtIgM4Pl3YfzXgJaqest/cUEe2HvCdkS46iW+qe8LQtKiwQt82MwUGeue8RQxaBj8424t65CZpb2t1Z1kmiXFXCM742teJ636DmWvZj/hUKId6Aspx7G4CtQogXPBIVuyEvMxHnG66jpoWXOXC3YzVtaOnq19xJYCISI0OwKMXArXsPOVjZgJBAHVaka2tV53EvE+XyGm8IIS55IiB2szy+UvWY4soGBOpJ82s/OSo/IwEnas1o7OhVOxS/JoRASWUjVs6O9+u1n0bC/RheLjU2HLMTpnCC8oDiigYsM8YiMiRQ7VB8Qr4c7lxSyaP53KnyWifqzD03Lla1ZKxSR/vtJ+YydeRlJuLIpVa093DxWHcxNV2HqamLl9ZwwpzEKUiJCeWLJzc7WKkUh101T3u/m2O1oJ4BcIyIvuepYNjI8jKU4rGfnvePWoPe6KBsBWihQrSrECklub642Izufi4e6y4llQ1YnKLU59SaMUsdQVlGYyMRXeBEpZ7FKQbETQni+mduVFzZgHlJEZgeHaZ2KD4lLyMRfYMWHLrQrHYofqmhoxcnrrRrtmU/3ig+s6zH930AzxDRkKxy/hQRLSaimQBARJHy8VNy/yoPxK4Zeh3h3rkJ+Pgc1z9zB3N3P8qq2zR7EpiMpTNjEBEccKMFylzL+rlq9XfToUESsuxQOoD7oAw53wFliY0qIhoCYAZwEMoy8Ltsqp0zF8nLTEQn1z9zi0/ONWHIIpDHw8udFhSgw91z43HwbCOX5HKDg5UNSIkJxZxEbRSHtefUKD6ZqLZal90AkA4gXS7BESvLIh10T6jatnJ2HNc/c5PiygbERwRjYXKU2qH4pLyMRDRf55JcrmatC5mXod26kJMaZi7nSPH8KA8IC1Lqn5VUNnD9MxfqH7Tgs3NNWDVXWyVkXOmeufHQ64i7+VzMWhdSq917AM+D8il5mYmoae3GxUauf+YqRy+3orNvkEfvTYIhLAg5qdE83NzFDlY2ar4upMcSFBEZ5CKEhURUNMpzCokoj4g2j7VNq1bLeRA8MdJ1iisaEBygw52ztVVCxtXyMxJxtr4TV9q4JJcrWCwCB8824u458ZquC+nQTz6RBQpHsAFArnXRQfuEY7MoYYl8nDfSNhfE4bOSokIwPzmSi8e6iBACB882YEV6HMKCAtQOx6dZW6DczecaJ+va0Xy9T9Pde4DjLSgzEf1sMgeSVc93yYdGACa7pyy12WYCkD3KNk1bPS8RZTVtaLnep3YoPu9C43XUtvZw954LGOOnwBgXzt18LlJS0QC9jnCPxutCOpqgNgDYQEQtk52wS0RGAK3WVpENg93j2FG22b/fZrlwYmlTk/9XWsjPTIQQwMfn/P9ndbdiOfF5tQZLyLjD6owEHDa1oLOXS3JNVkllA3JTo2EIC1I7FFU5Og9qr5wH9QyAn8vKEo9M8JiFQogtI2w3A7C/GzjSNvvYdgkhcoUQufHx/n+1kTUtEkmRIdzN5wIHKxswPzkSSVEhaofiF/IyEjEwJPA5V5WYlCtt3Thb36n57j3A+XlQu4QQMQDeBPAGER0hokWOvp6ICoUQO+T39t11RzHcYjICKB5lm6YREVZlJOCz803oGxxSOxyf1Xy9D8dqzXwScKGc1GhEhQZySa5J4rqQwyY0PEQIsR1Ajnx9ORG9SkSpY71GDnAoIqIyIiqDbBkRUbF8z70AjPJ5Bjkp+JZtE4nX3+RnJKKrfwiHTVxVYqI+OtsIIbRbQsYdAvQ6rJqnlOQa5JJcE1ZS2QBjfDiM8dqsHmHLoaFLRPQbKAnFKL8M1l3y3wIA64moSAjx7EjvIZNL2gjb822+3yG/LRlrm9YtT4tFaKAeBysbcPcc/+/WdIeSigZMjQpB1jRXDFBlVqszEvDWsTqU15g1PX9noq73DeJrUyu+vWKm2qF4BUdbUPkAogGUAfg3yCHjAKJlmaMYKHX6NhDRq26JlN0QEqjHnbPjcLCykatKTEDvwBA+v9CM1RkJmi0h4y53zYlHoJ74HukEfX6+Cf1DFqyex917gOMJqkoIUSDr8P1cLgF/TAjRbn2CbCFZW1J8Wepm+RmJqDP3oPJap9qh+JyvqlrQMzCE1dy953KRIYG4fVYsijlBTUhxZQMMYYHISY1WOxSvMNaKukvkqro/BpA31j0ma0ISQpigVDU3ujxSdpN75yWACDzvZAJKKhsQFqTHcuMtsxaYC+RlJMDU1AVTE5fkcsbgkAUfn23EqrkJCNBw9QhbY30KZgCXAGyVj01yHtRrRPRjIlol14FaAmUJDgCAbGkdd1vEDAAQHxGMJSkGTlBOEkLgYGUjVs6OQ0igXu1w/JK1ZcpVJZxTXmNGW/cAL/tiY6wVdS/JLr10KIMhNgAokt9vhTJooQ1AKZRExjwsLzMRJ6+0o769V+1QfMaZqx2o7+jl0XtulBIThnlJEXzx5KSSygYE6XW4iwc+3eBoOzJN3nfaIYTYIJNWNJSBEVuhDKJgHpZvvVI9yycCRxVXNIBI6SJl7rM6IwGl1W0wd/erHYrPKKlowLK0WEwJ5rqQVmPdg9pPRI8CSmvKfr8Qol3OVdrNa0KpIz1hClJjw3hipBMOnm1A9oxoxE0JVjsUv5aXkYghi8DH57ibzxFVTddhau5CPk/OvclYLahLkIMdiOh1ef9pv+39J8+EyEZDRMjLSMQXVS3o6htUOxyvd9Xcg9N1HTxD3wMWTTcgbkowrwDtIOtFJo8svdlY96C2CiF+IR9uh1KHrx02959sBk38yP2hspHkZSSif9DC9c8cYL0nUsA3od1OpyPkZSTgk3ONXJLLASWVDciaFolphlC1Q/EqjhaLvSTr8G0QQqQLIXRQlsKwHTTBVJA7U9Y/4xvS4yquaMCsuHCkcQkZj8jPVEpyfVXVonYoXq3leh/Kqtt44M4IHB5sT0TPE9ER62MhRLnNoInZ7gmPjSdQr8O9c+Px0dlGDFm4qsRoOnoHcNjUgvzMRK4e4SEr0uMQGqjni6dxfHyuCRahJHR2M0dX1P0tlG6+SzbblrhifSg2eXmZiWjt6sexmrbxn6xRn55rwsCQ4JOAB4UE6nHXnDiUVHBJrrGUVDQgKZLrQo7EmQULtwkhNlo3CCGOAdgLYBcR3euO4JhjrPXPDvBovlEVVzQgNjwI2TO4hIwn5Wcmob6jF6fq2sd/sgb1DgzhswtNXBdyFM7U07Bfoh1y4cFfANhx69OZp0SGBGKZMRYHztTzleoIBoYs+PhcI1bNS4BexycBT1o1LwE6Gl69mN3si4vN6O4fQkFWktqheCVHE1QJlFbUSA7Awdp7RFRoXf9phH3ZRFRlXS+KiIrk9jYiKiaibQ7GqkkFWUm43NKNi41c/8ze16ZWdPYOcveeCmLCg5A7M4YT1CgOnGlARHAA14UchaMJ6mnIpTSIKMJun8NVJOQChKOJEUKkCSFyAGwCsFNuXy+EyLdZF4qNwFpVgrv5blVcUY+QQB1WzuYSMmooyEzE2fpO1LZ2qx2KVxmyCJRUNuCeeQkICuDisCNxdJi5CcpSGgUAzER0VM5/ugDgHwE8P9lA7FbLNcpjAoCBiEZtoRHRZiIqJaLSpqamyYbhs5KiQrAoxYADZ+rVDsWrCCFQXNGAO9PjERrExWHVYG258sXTzcpr2tDS1c/z8sbgcNqWZY1iAGyEsnBhNGS1c5sJvZNGRJvtWloxAFqJaOdIz5fzs3KFELnx8dq+Qr4vKxEnrrTjWnuP2qF4jTNXO3C1vZdPAipKjQ3HnMQpKK7giydbB87UI0ivwz1ztX3eGovT7UohxF5ZZaJAfu12cUw3dRnKBGSG0nIrdPGx/EpBpnKjlWvzDTtQ0QAdgcsbqawgMwlHL7ehtYuLxwJKy/5ARQPuSI9FREig2uF4LUfnQUXJOnxD8qtF3o9a5MpgiMhg93gzEWW78hj+LD1hCozx4dyVYmP/6XosnRmDWC4Oq6o185Nu3HNhwPmG66hu6b5xUclG5mgLajeU0kY/h3If6hkoXW/lztThI6I8ALm2LSG7UX0xAFptHr8un1MIjDvIgkG5Uv2qqgXt3QNqh6I6U9N1nGvoxJr5fBJQW9a0SCQbQrH/NHfzAcD+M/UgAvIyuWU/FkcTVB6Ap4QQTwshDsputwIo96N2ENEjjryJvI8VbZtohBD5Nt+b5Nwq62OzLKm0Vwix3cFYNa0gKxGDvMwBAGD/GVkclueYqI6IcF9WEj6/0IzrXHkfByrqsSTFgISIELVD8WqOJqjWkTbKRPMLcLFYr7F4ugHxEcE4wDeksf9MPRZOj0IyV4j2CmvmJ6F/yIJPNH7xVCeXfeELp/E5M1F3yyj7DgDIdU04bLJ0OkJ+ZiI+OdeE3gHtLnNwrb0Hx2vNuI9PAl4jJzUacVOCsE/j3XzWbk4eWTo+RxNUEYClRLSPiFLt9uVjlBYWU8e6+VPR3T+ET89rd17YAdm9xwnKe+jlxdPHZxs1ffG073Q95iVFwMjLvozL4fWgoLSS4gGYiOiCHNXXAmWibpEbY2ROut0Yg+iwQHx46praoahm3+l6pCdMQXoCnwS8yX1ZSejqH8IXF7W5wGZjRy+OVrdi7fypaofiE5yZqGuSZYiWAtgFZXXdgwA2CCFecFN8bAIC9TrkZyaipFKbq5m2dvXjyOVWrOHWk9e5Iy0OEcEB2K/Riif7z9RDCGDtAv7ddETAaDuI6CiAUgDFAMqFEJcBZaFCAOUeiY5N2NoFU/F66RUcutCM1RpbqbOksgFDFsHde14oKECHVRkJKK5owOCQBQF6bdWg++BUPdLiwzGbW/YOGeu3oxxKa2kvgCo5QXe/XFn3ESLi1bW82Iq0OESEBOBDDd6Q3ne6HsmGUMxP5l9Rb7QmKwlt3QP4+pK2bl23XO/D15dasG7BVF77yUGjJighxBZZ404HYDaA70OpvZcP4A0AbbKixH4i+plnwmWOCgpQuvkOnKlH/6BF7XA8pr17AJ9faMK6BUl8EvBS98xNQFiQHu9r7B7pgYoGWAT4/pMTHK5mLifnbrVJWkuhLMNxCcB6dwbJJmbt/Kno6B3EV6YWtUPxmAMV9RgYEnhg4TS1Q2GjCA3SY3VGIvadrsfgkHYunj44dQ2psWHImGq/YhEbjVMdwEQ0k4hmAsq9KCHEbpm0ZrslOjYpK2fHITxIr6nRfO+dvIbp0aFYOD1K7VDYGO5fMBWtXf2auXgyd/fjq6oWrJ3P3XvOcKZY7FEAVVCGmQ/K7YvdGBubpJBA5Ur1gLwh7e/auvrxxcVm3L+QTwLe7p658QgP0uP9k9q4eCquaMCgRWAdj95zijPFYtMgV9YFYP3rf4GI9rkjMOYa6xYkobWrH4dN/n9D+kBFPQYtAg8s4O49bxcSqEd+ZiL2nanHgAYunj44dQ3JhlAsSOaWvTOcLRb7cyiLFVptAHCbMxXNmWfdMzcBU4ID8O6Jq2qH4nbvnbyGGTFhPHrPR9y/cBrM3QP4ssq/u/lau/rx+YVmPLCIW/bOcqZYrLDfKJdl/zdwsVivFRKoR0FmIj44fc2vJ+22dvXjy6oWPMDdez7jrjnKpN33T/r3xdMHp65h0CLw4CJu2TvL0QR1EMCzo+wrA2B05E2IqNBu/Sf7/W1EVExE2+xek0dEmx2Mldn5xuJp6OwdxKfn/Lc2377T9RiyCNy/kIfw+orgAD3ys5TRfP48FeKdE1eRFh+OzKncsnfWqAmKiJ6yebgNQAwRHQGwxO6p6wGYHDmYAwsOrhdC5AshdsgYrAsVlsjHeY4ch93szvQ4xIQH4R0/7uZ7/9RVzIrjk4CveWChMhXCX2vzXWvvwdHLrXhwUTK37CdgrBbULiI6T0SPCCHaodyHaodSWYKI6CgRXQCwCYCrFhM0EJFta2wphpOfCcAty7/LZeFLiai0qcl/WwiTEajXYd2CJJRUNqDLDxeLa+zsxVdVLbifZ+j7nDvT4xEZEuC3F0/vnbgGIYAHF3P33kSMlaDSAZwA8IZsOc2Uq9+mQVkbqgxKRYlcIcSbLoonBkArEe2Ujw12+2PtXyAnEOcKIXLj4+NdFIb/eXBRMnoHLCipbFA7FJd75/hVWATw8BI+CfiaoAAd7l84FfvP1KO73/8unt45cRULkqMwKy5c7VB80liljkxCiPVQltloB3BQDimPtJmg+7QQ4pirgpHJxgzALLv3zFCSFpuk3NRoTI0KwTvH/e9K9a1jdViQHIX0BJ6h74seWTId3f1Dflfh3NR0Hafq2vEQt54mbNxBErJiRD6G14MqJ6JXR1i4cFJkV519F95RDLeijFAqq7MJ0OkIDy6ahk/PN6Gtq1/tcFzmfEMnzlztwCNLktUOhU1Qbmo0pkeH4q1j/nXx9M6JqyACl92aBGfWgyqX60HdB6V4rImI/o8zVc3lIIdc6+AHuc2adF6Xj60DI/bKQRVG+TqDdbAEm5hvLJqGQYvwqwrnbx2rg15H+AYP4fVZOh3h4cXJOHShCY0dvWqH4xJCCLxz4ipumxmDpKgQtcPxWU4vxiKEKJGJagOAAihVzR2aqCtfG207mk+2ziCEMMskuFcIsd1m/w75uh3OxspuljUtEmnx4Xjr2BW1Q3EJi0Xgz8fqsHJ2HOIjgtUOh03CI9nJsAj4zWCJk1faYWrqwkOLuWU/GRMpFvsoEf0Yyqg+E5SyR5w8fAARoTAnBUcvt+FSc5fa4Uza4UstuNrey917fiAtfgoWTY/Cm+V1aofiEnvKahEcoMMDi3he3mSMNQ/qeSJ6TQ4nbyGiISjFYvdCSUj58qm7oNToYz7g0exk6AjYW1ardiiT9vaxOkwJDkBBJhfg9AcPL0lGxbUOnKvvVDuUSekdGMI7x69i7fwkRIYEqh2OTxurBbUdQA6ANgB7oJQzKgCQJoTQCSHShRAFcjTfzz0QK3OBxMgQ3D0nHm+U1WHIckv1Kp/ROzCED0/VY838JIQG6dUOh7nANxZNg15HeOuYb7eiDlQ0oKN3EOtzU9QOxeeNNczcPgntFkIcFEJc8mSAzPXW56agvqMXh3x49v6BigZ09g3iUe7e8xtxU4Jx95x4vH3Mty+e9pTWItkQiuXGW6ZtMic5PUiC+b7VGQkwhAXi9VLf7eZ79UgNkg2huJ1PAn5lfc501Hf04tPzjWqHMiF15h4cutiMx3KmQ6fjqiaTxQlKg4ID9Hh4cTKKzzTA3O17c6IuNXfhy6oWPH5bCvR8EvAreZmJiJsSjD9+7ZsXT2+WXYEQSqJlk8cJSqPW505H/5DFJ4f1vnq0BnodYQP38fudQL0OG3Kn46OzDahv9605UUII7C2/gmXGGKTEhKkdjl/gBKVRWdOikDk1EntKfWtOVP+gBXtLryAvIwEJkTwB0h99c+kMWAR8rgv6yKVWVLd084WTC3GC0rCNS1Nwqq4dJ2rNaofisOKKBrR09ePx22aoHQpzkxmxYVg5Ow6vHa31qcESr3xdg4iQAKyZz9MeXIUTlIY9mp2M8CA9XvrqstqhOOyPR6qRbAjFytlcud6fPX7bDNSZe/DZed9YQqexoxcfnLqGDbkpCAsKUDscv8EJSsMiQgLxWM50vHfiGlqu96kdzrguN3fhi4s8OEIL8q2DJY7UqB2KQ/54pAaDFoEnlrm0hrbmcYLSuCeXp6J/yIJXj3p/f/+rR2uh1xFPgNSAQL0O63On46OzjV4/WKJ/0IJXvq7BPXPjMZPXfXIpTlAal54QgTvT4/Dy4WoMDlnUDmdUPf1DePVoDfIyEpDIgyM04fGlM2ARAi8frlY7lDHtO1OPps4+/NUdM9UOxe9wgmJ4cnkqrrX3orjCe1fb3VtWC3P3ADatNKodCvOQGbFhuC8zCX84XO3Vq+2+9OVlpMaG4W6+L+pyHk1QRFRos/6T/T4DEWXL5xTZbG8jomIi2ua5SLVldUYikg2hXjtYYsgi8MKhS1gyw4Cc1Gi1w2EetOkuI9p7Brx2OsTpunaUVbfhiWWpXDnCDTyaoGzXgRrBBgC51ucQ0Wa5fb0QIp/Xg3IfvY7wxPJUHDa14mx9h9rh3KK4oh7VLd3YtNIIIj4JaElOajSyZxjwwiGTVw45f+nLywgN1PN9UTfxmi4+IcQuIcQu+dAIZa0pADAQ0aj9OnKp+FIiKm1q8o0hqd5oY24KwoL0+M0nVWqHcotdn5mQEhOK+7J4fokWbb7LiNrWHuw/410rQdeZe/D28ToU5kxHVCgvq+EOXpOgrGQyarVZ3j0GQCsR7Rzp+TKx5QohcuPjuQ94oqLDg/CtZal498RVr1rMsKy6FeU1Zjx1p5GHlmtUfmYSUmPDsOszE4TwnlbUrk+rIASw5W6+L+ouXpegABQKIbZYH8gEZAZgJqJC9cLyf0+tnIVAvQ7/5+OLaodyw+7PLiEqNBDrc7n4plbpdYSn7pyF47VmlFW3qR0OAGVi7p+O1uKx7OmYHs1199zFqxIUERVa7zXJARObiShb7bi0IiEiBI/fNgNvHatDbWu32uHgYmMn9lfU41vLZvDsfI0rzElBdFgg/ttLLp52fabcE/vBvWlqh+LXPD2KLw9Arm1LyDqqT+4rIqIyIiqD0rX3utxXCIw7yIK5wNa706Ajwm8+Vf9e1C/2n0d4UAC+u2KW2qEwlYUG6bHpLiM+PteEo5dbVY2l5XofXvm6Bg8tmobUWJ6Y606eHsVXIoSItk00Qoh8m31pQogc+VUihDALIcqFEHuFENs9GatWJUWFYH3udOwtvYJr7T2qxXGspg37ztRj00ojYqcEqxYH8x7fuWMWEiKCUfThWVXvRb1w6BJ6B4fwg3vTVYtBK7yqi495h613p8EihGoj+oQQKNp3FrHhQXhqJbeemCI0SI+/y5uN0uo2fHRWnRV3W7v68fsvL+P+BVORnjBFlRi0hBMUu0VKTBg2Lk3BK1/X4EJDp8eP//mFZhw2teJvVqUjPJjvPbFhG3JTMDM2DDv2nVNlXtQvD5xD76AFf7d6tsePrUWcoNiIflQwF+FBejz37hmPdqdYLErraXp0KB6/ndd8YjcL1Ovwo4K5ONfQiT8fr/PosU/XteOPR2rw5PJUzE6M8OixtYoTFBtRTHgQflQwF19cbPHoBMl3T17Fmasd+FHBHAQH6D12XOY77l8wFfOTI/HLA+fROzDkkWMKIfDcO2cQExaEv8+b45FjMk5QbAx/efsMzEuKwE/fq0RPv/tPBK1d/fjpexVYkByFBxclu/14zDfpdIRn12WgztyDXxef98gx/3z8Kkqr27BtzVyuGuFBnKDYqAL0Ojz3YBbqzD3Y+Zn7B0z85N0zaO8ZwI7ChVw1go3pjrQ4PH5bCnZ/bsKxGvdO3r3eN4iffVCJhdOjsD6Ha+55EicoNqZlxlg8sHAqfvNJFc67ccBEcUUD/nz8Kn54bzoypka67TjMfzyzLgOJkSHYtvck+gbd18L/+b6zaOzsw3MPZnHFcg/jBMXG9S8PZCIiJAA/fKXcLevytHcP4J/eOoV5SRH4wT08t4Q5JjIkED97dAEuNF7Hfx50T4WJD09dw0tfVeM7K2YiewYv9eJpnKDYuBIiQ/Af31yCi03X8c9vn3bpqD4hBH7y7hm0dPXjF+sXISiAfyWZ4+6dm4DHsqfjN59W4Xit2aXvXd3ShW17T2JRigHPrM1w6Xszx/DZgDlkRXoc/nbVbLxZXoc9Za5bPO43n1bhzWN1+Ot70zE/Ocpl78u0418eyERSZAg2/b7UZTUk+waH8MM/loMI+K/Hl/CFk0r4U2cO+9vVs3FHWiz+5c+ncbqufdLv99axK9ix7xweWjyNJz6yCYsKC8T/fGcp+gaG8O0Xj8Dc3T+p91Na9RU4XdeBX6xfhJQYrlauFk5QzGF6HeHfv7kY0WFB+Ivdhye19MEXF5uxbe9JLDfGYkfhQr75zCZldmIEdj+Zi9rWHmz6femE50dZLAL//PZp/PHrGmy524gCXiRTVZygmFMSIkLw+pblygKHL3yNzy84v4rxJ+casfUPZTDGTcFvn8jhCbnMJW43xuJXGxfh6OU2bH25zOmW1MCQBX//2nG88nUNtt6dhqfXzHNTpMxRnKCY01JiwrBn63Kkxobhe/9TireOXXFo4ET/oAX/3/sV+PaLR5EcHYoXv7OUJz0yl3pg4TT87JEF+OJiM9b9x+c4csmxpTnauvqx5Q9leOfEVWxfMw9Pr50HIm7Vq408WWdNruu0xbrExij7zQCMQohdo20bTW5urigtLXVpzGx07d0D+O5LR1FW3YZFKQZsu28uVqTH3fI8i0XgWK0ZP3n3DE5eaccTy1LxT/dnICSQW07MPU5eMeNv/3QMNa3d+Ot70/HE8pmIj7h12ZbO3gH8/4cu4YXPL6G7fxA/fXg+/vL2VBUi1jYiKhNC5N6y3dPrqhBR8UgJynZRQiLaDMAEwGC/TQhRMtp7c4LyvMEhC94sr8O/l5zH1fZe5KZGI2taJBKjQhA3JRgnr5hRXNGAho4+RIUGouixhVgzn/v1mftd7xvE//v2abx1rA5EQG5qNPIzExEaFICG9l7Ud/TiYGUD2roHsCYrCf9PwRzM4SKwqvCFBFUE4DUhRLlcXTcbQKz9NuuS8Dav2wxgMwDMmDEjp7q62v0/BLtF78AQ/nSkBq8drUWduQedvcqE3tBAPe6ZG4+CrESsmpfIXXrM4yqudmD/mXocqGhA5bUOAICOlPup85Mj8TerZmNRikHdIDVutATlTYvtGOwex46y7Say228XoLSg3BEYG19IoB7fWTEL35HLs3f3D6Kxow9JUSHclcdUlTktEpnTIvEP+XNQ394LIiBuSjDXe/QB3pSgzABiHNjGfEBYUABmxnnTrxdjQFJUiNohMCd40xnkKIZbTEYAxfKx/TbGGGMa4NFh5vI+Uq51QITcVgwoAyEAGOVzDEKIkpG2eTJexhhj6vH4IAl34lF8jDHme0YbJMETdRljjHklTlCMMca8EicoxhhjXsmv7kERUROAyc7UjQPQ7IJwfB1/DsP4sxjGn4WCP4dhrvgsUoUQ8fYb/SpBuQIRlY50s05r+HMYxp/FMP4sFPw5DHPnZ8FdfIwxxrwSJyjGGGNeiRPUrcZc0kND+HMYxp/FMP4sFPw5DHPbZ8H3oBhjjHklbkExxhjzSpygGGOMeSVvqmbuUeMtJe/MUvO+bqyflYgMUCrJGwEsFUJs93iAHuTo/zsRFfnzZ+HA30c2lN8Ja6Fnv8XnimHyZ90y0qKzNvvNcNFnockWlM3y8iXycZ4z+/2JAz/rBgC51pOQXMHYLzn6/y63Gz0Ymkc5+Dk8I38nYohIs5+FfGyS+00ycfutsS5G3HHe1GSCArAUgEl+b4KyvLwz+/3JmD+rEGKXzZWQ0ea5/mjc/3d5MvbnzwAY53OQFylHicgofz/8+fMY73eiFMAea4tSCFHuyeC8jMvPm1pNUAa7x/ZLyY+3358Y7B6P+LPKE3Orn6/JZbB7PNJnYfTzEzIw/ueQJre1EtFO2Q3srwx2j2/6LIQQZgA7AewBkOOZkLyWwe7xpM+bWk1QZoy9lPx4+/2JGY79rIVCiC1ujkVtZozxWRBRnp8naCszxv+dqJIn5zIAftvtCwd+JwCUCCHSAJhtF2PVIDNcfN7UaoIaaXl5Z/b7k3F/ViIqFELskN/7c3fneJ9FKxHlyZOQ0Y8/C0f+PqwMUE5M/mq8zyLbplvveWjnwnYkLj9vajJBjbaU/FjLz6sXrXuN91nI7UVEVEZEZfDjP0AHfi/K5bYY3Nqd4Tcc/PswWG+C+/PItfE+CwC7iGiz3L/Bnz8L4Mb5INe2pejO8yZXkmCMMeaVNNmCYowx5v04QTHGGPNKnKAYY4x5JU5QjDHGvBInKMYYY16JExRjbkZEBiIqtPnXpUNniShbTgEYbX8REW1z5TEZ8wQeZs6Ym8kyUVVQSuGYoBTfddncOpmcto/2nrIU0SUhRLSrjsmYJ3ALijH3s07qPAggz/pYTvAURFQs/91JRHuIqI2IiqwvltuEnCxtX7jVWqS0RLbQrK+vsqkubQZQ6s+V6Jl/4gTFmPtZ185ZPcr+YgC7oNS0M8nvt8mEUwSlKnQalMrZB+1em4vhCtKbARTK4+wFUGTzvHKbOBjzCZygGFOZrHO4Rz58HsBr8vsYKMnJiOGirAa79Zes3Yaw+df6XrYJqQp+vIYV80+coBjzfnuFENFCCJJftst9VEHWBZS10HIAlEBJZlU2z0uD/69jxfwMJyjGPGcihXaLAeQRkVHeo6qy218OpZsPsjvQ+vW83GaQz9PCQovMz3CCYszNZIunHMNdb868dgeUFpG1i2693f4SKMuAZENZOC9GPvcZKCP7zPKpeXI/Yz6Dh5kz5uPk6Lx8IcT6MfbnaGDBSeZnOEEx5geIyHbhPPt9RgCtNq0pxnwCJyjGGGNeie9BMcYY80qcoBhjjHklTlCMMca8EicoxhhjXokTFGOMMa/0fwHGPRGw32GQ7AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib\n",
    "matplotlib.rcParams['text.usetex'] = True\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "\n",
    "t = np.linspace(0.0, 1.0, 100)\n",
    "s = np.cos(4 * np.pi * t) + 2\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(6, 4), tight_layout=True)\n",
    "ax.plot(t, s)\n",
    "\n",
    "ax.set_xlabel(r'\\textbf{time (s)}')\n",
    "ax.set_ylabel('\\\\textit{Velocity (\\N{DEGREE SIGN}/sec)}', fontsize=16)\n",
    "ax.set_title(r'\\TeX\\ is Number $\\displaystyle\\sum_{n=1}^\\infty'\n",
    "             r'\\frac{-e^{i\\pi}}{2^n}$!', fontsize=16, color='r')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A more complex example.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "# interface tracking profiles\n",
    "N = 500\n",
    "delta = 0.6\n",
    "X = np.linspace(-1, 1, N)\n",
    "ax.plot(X, (1 - np.tanh(4 * X / delta)) / 2,    # phase field tanh profiles\n",
    "        X, (1.4 + np.tanh(4 * X / delta)) / 4, \"C2\",  # composition profile\n",
    "        X, X < 0, \"k--\")                        # sharp interface\n",
    "\n",
    "# legend\n",
    "ax.legend((\"phase field\", \"level set\", \"sharp interface\"),\n",
    "          shadow=True, loc=(0.01, 0.48), handlelength=1.5, fontsize=16)\n",
    "\n",
    "# the arrow\n",
    "ax.annotate(\"\", xy=(-delta / 2., 0.1), xytext=(delta / 2., 0.1),\n",
    "            arrowprops=dict(arrowstyle=\"<->\", connectionstyle=\"arc3\"))\n",
    "ax.text(0, 0.1, r\"$\\delta$\",\n",
    "        color=\"black\", fontsize=24,\n",
    "        horizontalalignment=\"center\", verticalalignment=\"center\",\n",
    "        bbox=dict(boxstyle=\"round\", fc=\"white\", ec=\"black\", pad=0.2))\n",
    "\n",
    "# Use tex in labels\n",
    "ax.set_xticks([-1, 0, 1])\n",
    "ax.set_xticklabels([\"$-1$\", r\"$\\pm 0$\", \"$+1$\"], color=\"k\", size=20)\n",
    "\n",
    "# Left Y-axis labels, combine math mode and text mode\n",
    "ax.set_ylabel(r\"\\bf{phase field} $\\phi$\", color=\"C0\", fontsize=20)\n",
    "ax.set_yticks([0, 0.5, 1])\n",
    "ax.set_yticklabels([r\"\\bf{0}\", r\"\\bf{.5}\", r\"\\bf{1}\"], color=\"k\", size=20)\n",
    "\n",
    "# Right Y-axis labels\n",
    "ax.text(1.02, 0.5, r\"\\bf{level set} $\\phi$\",\n",
    "        color=\"C2\", fontsize=20, rotation=90,\n",
    "        horizontalalignment=\"left\", verticalalignment=\"center\",\n",
    "        clip_on=False, transform=ax.transAxes)\n",
    "\n",
    "# Use multiline environment inside a `text`.\n",
    "# level set equations\n",
    "eq1 = (r\"\\begin{eqnarray*}\"\n",
    "       r\"|\\nabla\\phi| &=& 1,\\\\\"\n",
    "       r\"\\frac{\\partial \\phi}{\\partial t} + U|\\nabla \\phi| &=& 0 \"\n",
    "       r\"\\end{eqnarray*}\")\n",
    "ax.text(1, 0.9, eq1, color=\"C2\", fontsize=18,\n",
    "        horizontalalignment=\"right\", verticalalignment=\"top\")\n",
    "\n",
    "# phase field equations\n",
    "eq2 = (r\"\\begin{eqnarray*}\"\n",
    "       r\"\\mathcal{F} &=& \\int f\\left( \\phi, c \\right) dV, \\\\ \"\n",
    "       r\"\\frac{ \\partial \\phi } { \\partial t } &=& -M_{ \\phi } \"\n",
    "       r\"\\frac{ \\delta \\mathcal{F} } { \\delta \\phi }\"\n",
    "       r\"\\end{eqnarray*}\")\n",
    "ax.text(0.18, 0.18, eq2, color=\"C0\", fontsize=16)\n",
    "\n",
    "ax.text(-1, .30, r\"gamma: $\\gamma$\", color=\"r\", fontsize=20)\n",
    "ax.text(-1, .18, r\"Omega: $\\Omega$\", color=\"b\", fontsize=20)\n",
    "\n",
    "plt.show()"
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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